Elliptic Curve Diffie Hellman

Elliptic Curve Diffie Hellman example.

Prerequisites

• bun

Install dependencies

bun install

Run the script

bun run src/index.ts

SECP256K1 parameters for reference

Param	Value
p	115792089237316195423570985008687907853269984665640564039457584007908834671663
a	0
b	7
Gx	55066263022277343669578718895168534326250603453777594175500187360389116729240
Gy	32670510020758816978083085130507043184471273380659243275938904335757337482424
n	115792089237316195423570985008687907852837564279074904382605163141518161494337
h	1

Normal Diffie Hellman

• $p$ The order of the field
• $G$ Generator number
• $a$ Alice Private key
• $A$ Alice Public key
• $b$ Bob Private key
• $B$ Bob Public key
• $S$ Shared secret

Alice	Attacker sees	Bob
$a \in [1, p-1]$	$p,g$	$b \in [1,p-1]$
$A=g^a\mod{p}$	$A,B$	$B=g^b\mod{p}$
$S=B^a\mod{p}=(g^b)^a \mod{p}$		$S=A^b\mod{p}=(g^a)^b\mod{p}$

Elliptic Curve Diffie Hellman

• $n$ The key size
• $G$ Generator (Point) of the curve
• $a$ Alice Private key
• $A$ Alice Public key (Point)
• $b$ Bob Private key
• $B$ Bob Public key (Point)
• $(S_x,S_y)$ Shared secret point (usually only $S_x$ is used)

Alice	Attacker sees	Bob
$a \in [1,n-1]$	$n, G$	$b \in [1, n - 1]$
$A=a\cdot G$	$A, B$	$B = b \cdot G$
$(S_x,S_y)=a \cdot B=a\cdot (b \cdot G)$		$(S_x,S_y)=b \cdot A = b \cdot (a \cdot G)$